viervorm

Viervorm is the Dutch term for a four-form, a concept in differential geometry. It denotes a differential form of degree four on a differentiable manifold.

A viervorm is an antisymmetric covariant tensor of rank four. Locally it can be expressed as a sum of smooth functions times wedge products of four differentials, for example ω = ∑ f_{ijkl} dx^i ∧ dx^j ∧ dx^k ∧ dx^l. Because of the antisymmetry of the wedge product, a viervorm vanishes if any two indices coincide.

The space of viervormen on a manifold M is the fourth exterior power of the cotangent bundle, denoted Ω^4(M). On an oriented 4-manifold, a top-degree form can be integrated, and the exterior derivative d maps Ω^k(M) to Ω^{k+1}(M) with the property d^2 = 0. A viervorm is therefore a candidate for defining a volume element when it is nowhere vanishing and compatible with the orientation.

A canonical example occurs on R^4 with standard coordinates (x^1, x^2, x^3, x^4), where the volume form dx^1 ∧ dx^2 ∧ dx^3 ∧ dx^4 is a basic viervorm. Such a form defines oriented volume locally and serves as a reference against which other 4-forms can be measured.

Viervormen have applications in geometry and physics. They are used to express volumes and flux integrals in four dimensions and appear in theories that employ differential forms, such as general relativity and various field theories. They are related to other forms through the exterior algebra and the Hodge star operation, which connects 4-forms to 0-forms in four-dimensional settings.